Abstract
Embedding geometries in structured grids allows a simple treatment of complex objects in fluid simulations. Various methods for embedding geometries are available. The commonly used Brinkman-volume-penalization models geometries as porous media, and approximates a solid object in the limit of vanishing porosity. In its simplest form, the momentum equations are augmented by a term penalizing the fluid velocity, yielding good results in many applications. However, it induces numerical stiffness, especially if high-pressure gradients need to be balanced. Here, we focus on the effect of the reduced effective volume (commonly called porosity) of the porous medium. An approach is derived, which allows reducing the flux through objects to practically zero with little increase of numerical stiffness. Also, non-slip boundary conditions and adiabatic boundary conditions are easily constructed. The porosity terms allow keeping the skew symmetry of the underlying numerical scheme, by which the numerical stability is improved. Furthermore, very good conservation of mass and energy in the non-penalized domain can be achieved, for which the boundary smoothing introduces a small ambiguity in its definition. The scheme is tested for acoustic scenarios, for near incompressible and strongly compressible flows.
Highlights
Most flows in technical applications are defined by the geometrical properties of the enclosing or the contained objects, which are often of complex shape
Journal of Scientific Computing (2022) 90:86 commonly grids are constructed, where cell boundaries or grid lines are aligned with the boundaries of the objects. This simplifies the implementation of boundary conditions, as these conditions can be enforced at specific cell boundaries or grid lines
Liu and Vasilyev [40] treat compressible flows by a Brinkman penalization method popularizing the approach in the compressible regime
Summary
Journal of Scientific Computing (2022) 90:86 commonly grids are constructed, where cell boundaries or grid lines are aligned with the boundaries of the objects. Liu and Vasilyev [40] treat compressible flows by a Brinkman penalization method popularizing the approach in the compressible regime They include both, the linear friction relative to the object (Darcy’s law), and the effective volume fraction remaining for the fluid φ, which is often called porosity. The derived equations were presented before by Liu and Vasilyev [40], as the fundamental equations for flow through porous material by Darcy, but where turned down following arguments in Nield and Bejan [44] and Beck [7], suggesting structural problems Here no such structural problems appear, instead, it is found that the volume fraction and Darcy term can be chosen largely independently, resulting in a substantial extension of possible boundary conditions for the Brinkman penalization method.
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